Szpiro's conjecture: Difference between revisions
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In [[number theory]], '''Szpiro's conjecture''' concerns a relationship between the [[conductor of an elliptic curve|conductor]] and the [[discriminant of an elliptic curve|discriminant]] of an [[elliptic curve]]. In a general form, it is equivalent to the well-known [[ABC conjecture]]. It is named for [[Lucien Szpiro]] who formulated it in the 1980s. | In [[number theory]], '''Szpiro's conjecture''' concerns a relationship between the [[conductor of an elliptic curve|conductor]] and the [[discriminant of an elliptic curve|discriminant]] of an [[elliptic curve]]. In a general form, it is equivalent to the well-known [[ABC conjecture]]. It is named for [[Lucien Szpiro]] who formulated it in the 1980s. | ||
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:<math> \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. \, </math> | :<math> \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. \, </math> | ||
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Latest revision as of 11:01, 24 October 2024
In number theory, Szpiro's conjecture concerns a relationship between the conductor and the discriminant of an elliptic curve. In a general form, it is equivalent to the well-known ABC conjecture. It is named for Lucien Szpiro who formulated it in the 1980s.
The conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with minimal discriminant Δ and conductor f, we have
The modified Szpiro conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with invariants c4, c6 and conductor f, we have
